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Systems I
Bits and Bytes
Topics


Why bits?
Representing information as bits
 Binary/Hexadecimal
 Byte representations
» numbers
» characters and strings
» Instructions

Bit-level manipulations
 Boolean algebra
 Expressing in C
Why Donʼt Computers Use Base 10?
Base 10 Number Representation

Thatʼs why fingers are known as “digits”

Natural representation for financial transactions
 Floating point number cannot exactly represent $1.20

Even carries through in scientific notation
 1.5213 X 104
Implementing Electronically

Hard to store
 ENIAC (First electronic computer) used 10 vacuum tubes / digit

Hard to transmit
 Need high precision to encode 10 signal levels on single wire

Messy to implement digital logic functions
 Addition, multiplication, etc.
2
Binary Representations
Base 2 Number Representation

Represent 1521310 as 111011011011012
Represent 1.2010 as 1.0011001100110011[0011]…2

Represent 1.5213 X 104 as 1.11011011011012 X 213

Electronic Implementation


Easy to store with bistable elements
Reliably transmitted on noisy and inaccurate wires
0
1
0
3.3V
2.8V
0.5V
0.0V
 Straightforward implementation of arithmetic functions
3
Encoding Byte Values
Byte = 8 bits

Binary
000000002
to
111111112

Decimal:
010
to
25510

Hexadecimal
0016
to
FF16
 Base 16 number representation
 Use characters ʻ0ʼ to ʻ9ʼ and ʻAʼ to ʻFʼ
 Write FA1D37B16 in C as 0xFA1D37B
» Or 0xfa1d37b
al y
im ar
c
x
n
He De Bi
0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111
4
Machine Words
Machine Has “Word Size”

Nominal size of integer-valued data
 Including addresses

Most current machines are 32 bits (4 bytes)
 Limits addresses to 4GB
 Becoming too small for memory-intensive applications

High-end systems are 64 bits (8 bytes)
 Potentially address ≈ 1.8 X 1019 bytes

Machines support multiple data formats
 Fractions or multiples of word size
 Always integral number of bytes
5
Word-Oriented Memory
Organization
32-bit 64-bit
Words Words
Addresses Specify Byte
Locations


Address of first byte in
word
Addresses of successive
words differ by 4 (32-bit) or
8 (64-bit)
Addr
=
0000
??
Addr
=
0000
??
Addr
=
0004
??
Addr
=
0008
??
Addr
=
0012
??
Addr
=
0008
??
Bytes Addr.
0000
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
6
Data Representations
Sizes of C Objects (in Bytes)

C Data Type
 int
 long int
 char
 short
 float
 double
 long double
 char *
Typical 32-bit
Intel IA32
4
4
1
2
4
8
8
4
4
4
1
2
4
8
10/12
4
» Or any other pointer
7
Byte Ordering
How should bytes within multi-byte word be ordered in
memory?
Conventions

Sunʼs, Macʼs are “Big Endian” machines
 Least significant byte has highest address

Alphas, PCʼs are “Little Endian” machines
 Least significant byte has lowest address
8
Byte Ordering Example
Big Endian

Least significant byte has highest address
Little Endian

Least significant byte has lowest address
Example

Variable x has 4-byte representation 0x01234567

Address given by &x is 0x100
Big Endian
0x100 0x101 0x102 0x103
01
Little Endian
23
45
67
0x100 0x101 0x102 0x103
67
45
23
01
9
Representing Integers
int A = 15213;
int B = -15213;
long int C = 15213;
Linux/Alpha A
6D
3B
00
00
Linux/Alpha B
93
C4
FF
FF
Decimal: 15213
Binary:
0011 1011 0110 1101
3
Hex:
B
6
D
Sun A
Linux C
Alpha C
Sun C
00
00
3B
6D
6D
3B
00
00
6D
3B
00
00
00
00
00
00
00
00
3B
6D
Sun B
FF
FF
C4
93
Twoʼs complement representation
(Covered next lecture)
10
Representing Pointers (addresses)
Alpha P
int B = -15213;
int *P = &B;
A0
FC
FF
FF
01
00
00
00
Alpha Address
1
Hex:
Binary:
Sun P
EF
FF
FB
2C
F
F
F
F
F
C
A
0
0001 1111 1111 1111 1111 1111 1100 1010 0000
Sun Address
Hex:
Binary:
E
F
F
F
F
B
2
C
1110 1111 1111 1111 1111 1011 0010 1100
Linux P
Linux Address
Hex:
Binary:
B
F
F
F
F
8
D
4
1011 1111 1111 1111 1111 1000 1101 0100
D4
F8
FF
BF
Different compilers & machines assign different locations to objects
11
Representing Floats
Float F = 15213.0;
Linux/Alpha F
00
B4
6D
46
Sun F
46
6D
B4
00
IEEE Single Precision Floating Point Representation
Hex:
Binary:
15213:
4
6
6
D
B
4
0
0
0100 0110 0110 1101 1011 0100 0000 0000
1110 1101 1011 01
Not same as integer representation, but consistent across machines
Can see some relation to integer representation, but not obvious
12
Representing Strings
Strings in C
char S[6] = "15213";

Represented by array of characters

Each character encoded in ASCII format
 Standard 7-bit encoding of character set
 Other encodings exist, but uncommon
 Character “0” has code 0x30
» Digit i has code 0x30+i

String should be null-terminated
 Final character = 0
Linux/Alpha S Sun S
31
35
32
31
33
00
31
35
32
31
33
00
Compatibility

Byte ordering not an issue
 Data are single byte quantities

Text files generally platform independent
 Except for different conventions of line termination character(s)!
13
Machine-Level Code Representation
Encode Program as Sequence of Instructions

Each simple operation
 Arithmetic operation
 Read or write memory
 Conditional branch

Instructions encoded as bytes
 Alphaʼs, Sunʼs, Macʼs use 4 byte instructions
» Reduced Instruction Set Computer (RISC)
 PCʼs use variable length instructions
» Complex Instruction Set Computer (CISC)

Different instruction types and encodings for different
machines
 Most code not binary compatible
Programs are Byte Sequences Too!
14
Representing Instructions
int sum(int x, int y)
{
return x+y;
}

For this example, Alpha &
Sun use two 4-byte
instructions
 Use differing numbers of
instructions in other cases

PC uses 7 instructions with
lengths 1, 2, and 3 bytes
 Same for NT and for Linux
 NT / Linux not fully binary
compatible
Alpha sum
00
00
30
42
01
80
FA
6B
Sun sum
PC sum
81
C3
E0
08
90
02
00
09
55
89
E5
8B
45
0C
03
45
08
89
EC
5D
C3
Different machines use totally different instructions and encodings
15
Boolean Algebra
Developed by George Boole in 19th Century

Algebraic representation of logic
 Encode “True” as 1 and “False” as 0
And

Not

Or
A&B = 1 when both A=1 and
B=1
& 0 1
0 0 0
1 0 1
~A = 1 when A=0
~
0 1
1 0

A|B = 1 when either A=1 or
B=1
| 0 1
0 0 1
1 1 1
Exclusive-Or (Xor)

A^B = 1 when either A=1 or
B=1, but not both
^ 0 1
0 0 1
1 1 0
16
Application of Boolean Algebra
Applied to Digital Systems by Claude Shannon

1937 MIT Masterʼs Thesis

Reason about networks of relay switches
 Encode closed switch as 1, open switch as 0
A&~B
A
Connection when
~B
A&~B | ~A&B
~A
B
~A&B
= A^B
17
Integer Algebra
Integer Arithmetic

〈Z, +, *, –, 0, 1〉 forms a “ring”

Addition is “sum” operation

Multiplication is “product” operation

– is additive inverse

0 is identity for sum

1 is identity for product
18
Boolean Algebra
Boolean Algebra

〈{0,1}, |, &, ~, 0, 1〉 forms a “Boolean algebra”

Or is “sum” operation

And is “product” operation

~ is “complement” operation (not additive inverse)

0 is identity for sum

1 is identity for product
19
Boolean Algebra ≈ Integer Ring






Commutativity
A|B = B|A
A&B = B&A
Associativity
(A | B) | C = A | (B | C)
(A & B) & C = A & (B & C)
Product distributes over sum
A & (B | C) = (A & B) | (A & C)
Sum and product identities
A|0 = A
A&1 = A
Zero is product annihilator
A&0 = 0
Cancellation of negation
~ (~ A) = A
A+B = B+A
A*B = B*A
(A + B) + C = A + (B + C)
(A * B) * C = A * (B * C)
A * (B + C) = A * B + B * C
A+0 = A
A*1 =A
A*0 = 0
– (– A) = A
20
Boolean Algebra ≠ Integer Ring


Boolean: Sum distributes over product
A | (B & C) = (A | B) & (A | C) A + (B * C) ≠ (A + B) * (B + C)
Boolean: Idempotency
A|A = A
A +A≠A
 “A is true” or “A is true” = “A is true”

A&A = A
Boolean: Absorption
A | (A & B) = A
A *A≠A
A + (A * B) ≠ A
 “A is true” or “A is true and B is true” = “A is true”

A & (A | B) = A
Boolean: Laws of Complements
A | ~A = 1
A * (A + B) ≠ A
A + –A ≠ 1
 “A is true” or “A is false”

Ring: Every element has additive inverse
A | ~A ≠ 0
A + –A = 0
21
Boolean Ring
Properties of & and ^

〈{0,1}, ^, &, Ι, 0, 1〉

Identical to integers mod 2
Ι is identity operation: Ι (A) = A

A^A=0
Property
Boolean Ring

Commutative sum
A^B = B^A

Commutative product
A&B = B&A

Associative sum
(A ^ B) ^ C = A ^ (B ^ C)

Associative product
(A & B) & C = A & (B & C)

Prod. over sum
A & (B ^ C) = (A & B) ^ (B & C)

0 is sum identity
A^0 = A

1 is prod. identity
A&1 = A

0 is product annihilator
A&0=0

Additive inverse
A^A = 0
22
Relations Between Operations
DeMorganʼs Laws

Express & in terms of |, and vice-versa
 A & B = ~(~A | ~B)
» A and B are true if and only if neither A nor B is false
 A | B = ~(~A & ~B)
» A or B are true if and only if A and B are not both false
Exclusive-Or using Inclusive Or
 A ^ B = (~A & B) | (A & ~B)
» Exactly one of A and B is true
 A ^ B = (A | B) & ~(A & B)
» Either A is true, or B is true, but not both
23
General Boolean Algebras
Operate on Bit Vectors

Operations applied bitwise
01101001
& 01010101
01000001
01000001
01101001
| 01010101
01111101
01111101
01101001
^ 01010101
00111100
00111100
~ 01010101
10101010
10101010
All of the Properties of Boolean Algebra Apply
24
Representing & Manipulating Sets
Representation


Width w bit vector represents subsets of {0, …, w–1}
aj = 1 if j ∈ A
01101001
{ 0, 3, 5, 6 }
76543210
01010101
76543210
{ 0, 2, 4, 6 }
Operations




&
|
^
~
Intersection
Union
Symmetric difference
Complement
01000001
01111101
00111100
10101010
{ 0, 6 }
{ 0, 2, 3, 4, 5, 6 }
{ 2, 3, 4, 5 }
{ 1, 3, 5, 7 }
25
Bit-Level Operations in C
Operations &, |, ~, ^ Available in C

Apply to any “integral” data type
 long, int, short, char

View arguments as bit vectors

Arguments applied bit-wise
Examples (Char data type)

~0x41 -->
~010000012

~0x00 -->
~000000002

0x69 & 0x55
0xBE
--> 101111102
0xFF
--> 111111112
-->
0x41
011010012 & 010101012 --> 010000012

0x69 | 0x55
-->
0x7D
011010012 | 010101012 --> 011111012
26
Contrast: Logic Operations in C
Contrast to Logical Operators

&&, ||, !
 View 0 as “False”
 Anything nonzero as “True”
 Always return 0 or 1
 Early termination
Examples (char data type)

!0x41
-->
0x00

!0x00
-->
0x01

!!0x41 -->
0x01

0x69 && 0x55
-->
0x01

0x69 || 0x55
-->
0x01

p && *p (avoids null pointer access)
27
Shift Operations
Left Shift:

x << y
Shift bit-vector x left y positions
 Throw away extra bits on left
 Fill with 0ʼs on right
Right Shift:

x >> y
Shift bit-vector x right y
positions
 Throw away extra bits on right

Logical shift
 Fill with 0ʼs on left

Arithmetic shift
Argument x 01100010
<< 3
00010000
Log. >> 2
00011000
Arith. >> 2 00011000
Argument x 10100010
<< 3
00010000
Log. >> 2
00101000
Arith. >> 2 11101000
 Replicate most significant bit on
right
 Useful with twoʼs complement
integer representation
28
Cool Stuff with Xor

Bitwise Xor is form
of addition

With extra property
that every value is
its own additive
inverse
void funny(int *x, int *y)
{
*x = *x ^ *y;
/* #1 */
*y = *x ^ *y;
/* #2 */
*x = *x ^ *y;
/* #3 */
}
A^A=0
*x
*y
Begin
A
B
1
A^B
B
2
A^B
(A^B)^B = A
3
(A^B)^A = B
A
End
B
A
29
Main Points
Itʼs All About Bits & Bytes

Numbers

Programs

Text
Different Machines Follow Different Conventions

Word size

Byte ordering

Representations
Boolean Algebra is Mathematical Basis

Basic form encodes “false” as 0, “true” as 1

General form like bit-level operations in C
 Good for representing & manipulating sets
30
Reading Byte-Reversed Listings
Disassembly

Text representation of binary machine code

Generated by program that reads the machine code
Example Fragment
Address
8048365:
8048366:
804836c:
Instruction Code
5b
81 c3 ab 12 00 00
83 bb 28 00 00 00 00
Assembly Rendition
pop
%ebx
add
$0x12ab,%ebx
cmpl
$0x0,0x28(%ebx)
Deciphering Numbers
0x12ab

Value:

Pad to 4 bytes:
0x000012ab

Split into bytes:
00 00 12 ab

Reverse:
ab 12 00 00
31
Examining Data Representations
Code to Print Byte Representation of Data

Casting pointer to unsigned char * creates byte array
typedef unsigned char *pointer;
void show_bytes(pointer start, int len)
{
int i;
for (i = 0; i < len; i++)
printf("0x%p\t0x%.2x\n",
start+i, start[i]);
printf("\n");
}
Printf directives:
%p: Print pointer
%x: Print Hexadecimal
32
show_bytes Execution Example
int a = 15213;
printf("int a = 15213;\n");
show_bytes((pointer) &a, sizeof(int));
Result (Linux):
int a = 15213;
0x11ffffcb8
0x6d
0x11ffffcb9
0x3b
0x11ffffcba
0x00
0x11ffffcbb
0x00
33